A no regular curve.

Question

How can I prove that there is not a parametrization $\alpha$ of the set $\{(x,|x|) : -1\leq x \leq 1\}$ such that $\alpha $ is regular curve?

Thanks a lot!

Answer 1

Here's a sketch: suppose $\gamma(t)$ is such a parameterization; then $\gamma(a) = (0,0)$ for some $a$. If $\gamma'(a)$ is parallel to $(1,1)$, apply Taylor's theorem to $\gamma(a-h)$ for sufficiently small $h>0$ to get a contradiction; otherwise, apply it to $\gamma(a+h)$.

The intuition here is that if $\gamma$ is a regular curve, then it has to have a good linear approximation at every point along the curve. Look at the linear approximation at the origin: if it has slope 1, then it can't be a good approximation to the left; if it has slope -1, it can't be good to the right (and if it has some other slope, it's bad in both directions!)